(a) Let ψ_0(q) be the wavefunction corresponding to the vacuum state |0⟩ is the representation of the position operator

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(a) Let ψ_0(q) be the wavefunction corresponding to the vacuum state |0⟩ is the representation of the position operator ˆq. Use the annihilation operator expression in quadrature terms ˆq, ˆp to obtain a differential equation for ψ_0(q) and solve it (thereby showing that the zero-photon vacuum state does indeed exist and it is a well-behaved state). Plot |ψ_0(q)|^2 and deduce that, even in the state vacuum, the quadratures show fluctuations (dispersion). Explain why it was
quadratures are expected to fluctuate.
(b) Construct a coherent state with an expected value of the number of photons equal to 5.
Find the probability that the state has 10 photons, the probability that it has
5 photons and the probability that it has 0 photons. Graph the probability of the number
of photons in the interval [0, 20] photons.
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